Wold Decomposition Theorem 1

14.384 Time Series Analysis, Fall 2007Professor Anna Mikusheva

Paul Schrimpf, scribeSeptember 25, 2007

Corrected October, 2012 Lecture 10

Introduction to VARs

Wold Decomposition Theorem

Theorem 1 (Wold decomposition). Let yt be a 2nd order stationary process with Eyt = 0. Then yt couldbe written as yt = vt + c(L)et, where

1. et: Eet = 0, Eetet = , and et span{yt, yt1, ...} = It ( i.e. et are fundamental)2. Eetes = 0 for all t = s

3.j=0 cj2 0. This implies that It = I e t1 { t}.We now verifty that et satisfies conditions 1-4.

1. Eyt = 0 t implies that Eet = 0.Also, Var(et) = E(yt a(L)yt)2 is just a function of the covariances of yt and aj , so Var(et) = 2 isconstant.

2. Eetytj = 0 j 1, and etj = ytj a(L)ytj implies Eetetj = 0.3. By definition of et,

yt =et +

k

akytk

=1

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6

Discussion 2

Repeatedly substituting for ytk gives

yt =et + akytk

k

=1

=et + a1(et1 +

akyt 1 k) +

a y k tk

k=1 k=2

=et + a1et 1 +

a kytk1

k=1

=et + a1e 1t1 + vt...k

=

c kjetj + vtj=0

where vkt span{ytk1, ytk2, ...}. Consider:k k

E(y k c e )2 =Ey2 2t j t j t +

cj

2 2

cjE[y e t tj ]j=0 j=0 j=0

k k

=Ey2t +

c2j2

j

2=0

c2j

2

j=0

k

=Ey2t

c2j2

j=0

where we used the fact that [ k E yte kt j ] = E[(l=0 clet l + vt )e 2t ] = c . Now we know that E(y j l t2k 2 j=0 cjet j) 0.Therefore, it must be that k 2=0 cj Eytj 2 and j=0 c2 j .

This implies that, j=0 cjet j is well-defined.

4. Finally, vt = yt j=0 cjetj I .

Discussion

Definition 3. If vt = 0, the yt is called linearly regular.

Wold decomposition (an MA representation satisfying conditions (1)-(4) of the theorem)is unique.We put unique in quotes, because it is only unique up to a rotation of et. Let R be a rotation. et = Rut,then yt = c(L)Rut = c(L)ut, where cj = cjR. What is really unique is the space spanned by {et}.Structural VARs are about how we pick a particular rotation of the, and then attach interpretationsto the components of et.

Its true that any 2nd order stationary process has an MA representation, but that doesnt mean thatits always a good idea to study the MA representation.

P (y = 1 y = 1) = pExample 4. Two-state Markov chain:

{t | t1 . This is 2nd order stationary and

P (yt = 0|yt 1 = 0) = q

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Discussion 3

has the following MA representation:

E(yt|yt1) ={p yt1 = 11 q yt1 = 0

=1 q + yt1(q 1 + p)so

yt =1 q + yt1(q 1 + p) + et1

= q

+

(p+ q j2 p q

=0

1) etjj

where 1 p prob p | yt = 11prob 1=p p | yt 1 = 1

et(1 q) prob q | yt1 = 0

q prob 1 q | yt1 = 0The MA representation is not the best description of the dynamics in this case. This example is simple,but the point is more general. There are macro models with simple state-space representations andcomplicated MA representations.

There are many MA representations. The Wold representation is only unique up to rotation when yourequire the MA to be fundamental.

Example 5. Suppose yt = et + e 2t 1, Var(et) = . Then = (1 + 2)2, = 2, = 0 j > 1.1

0 1 j Consider yt = t+ 2 2t with Var1 t = . It is easy to see that this process has the same covariances.However, only the first representation is fundamental.

yt =(1 + L)et

e 1t =(1 + L) yt =

()jytjj=0

For the second process, first define F = L1.

yt =(F + 1)t1

jt 1 =

() F jy tj=0

Remark 6. It is possible that your model has a non-fundamental MA representation. A fundamentalrepresentation requires that your shocks are spanned by the set of variables in your VAR. For example,if youre looking at shocks to money supply in a VAR with money and output, it is very likely thatthe Fed looks at variables other than output and money and when choosing policy, so shocks to themoney supply may not be fundamental. One solution might be to include more variables in your VAR.Relatedly, you might use a factor-augmented VAR.

Remark 7. Relation to spectrum decomposition: there are three ways to represent a process: the Wolddecomposition, the covariances, and the spectrum. There is a 1-1 mapping among these representations, inthe following sense: there is a one-to-one mapping between the spectrum and the set of auto-covariances;there is also a 1-1 mapping between autocovariances and the set of coefficients of the fundamental MArepresentation.

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14.384 Time Series AnalysisFall 2013

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Wold Decomposition TheoremDiscussion